Diamond on successors of singulars

ESI workshop on large cardinals

and descriptive set theory

18-Jun-09, Vienna

Assaf Rinot

Tel-Aviv University

http://www.tau.ac.il/∼rinot

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Diamond on successor cardinals

Definition (Jensen, ‘72). For an infinite cardinal, λ,

and a stationary set S ⊆ λ+, ♦(S) asserts the existence

of a sequence 〈Aα | α ∈ S〉 such that {α ∈ S | A∩α = Aα}is stationary for all A ⊆ λ+.

Theorem (Jensen, ’72). In Godel’s constructible uni-

verse, ♦(S) holds for every stationary S ⊆ λ+ and every

infinite cardinal, λ.

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Notation and conventions

Let Eλ+κ := {δ < λ+ | cf(δ) = κ},

and Eλ+

6=κ := {δ < λ+ | cf(δ) 6= κ}.

We shall say that S ⊆ λ+ reflects iff the following set

is stationary:

Tr(S) := {γ < λ+ | cf(γ) > ω, S ∩ γ is stationary}.

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Diamond vs. GCH

Observation. For S ⊆ λ+, ♦(S)⇒ ♦(λ+)⇒ 2λ = λ+.

Theorem (Jensen, ‘74). CH 6⇒ ♦(ℵ1).

Theorem (Gregory, ‘76). GCH⇒ ♦(ℵ2). Moreover:

GCH entails ♦(S) for every stationary S ⊆ Eℵ2ℵ0.

Theorem (Shelah, ‘78). For uncountable cardinal λ:

GCH entails ♦(S) for every stationary S ⊆ Eλ+

6=cf(λ).

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Successors of regulars

Theorem (Shelah, ‘80). For every regular uncountable

cardinal, λ:

GCH + ¬♦(Eλ+

cf(λ)) is consistent.

Thus, the possible behaviors of diamond at succes-

sors of regulars, in the presence of GCH, are well-

understood.

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Successors of singulars

Theorem (Shelah, ‘84). For every singular cardinal, λ,

for some non-reflecting stationary set S ⊆ Eλ+

cf(λ):

GCH + ¬♦(S) is consistent.

and in the other direction:

Theorem (Shelah, ‘84). For every singular cardinal, λ:

(GCH and �∗λ) entails ♦(S) for every S ⊆ Eλ+

cf(λ) that

reflects.

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Questions

For 25-30 years, the following questions remained open:

Question 1. Could GCH be replaced with “2λ = λ+”

in the above combinatorial theorems?

Question 2. To what extent can �∗λ be weakened?

Question 3. Can �∗λ be completely eliminated?

put differently, can GCH hold while ♦(S) fails for a set

S ⊆ Eλ+

cf(λ) that reflects?

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Status

Question 1 has recently been answered in the affirma-

tive(!)

Theorem (Shelah, 2007). For uncountable cardinal, λ:

2λ = λ+ entails ♦(S) for every stationary S ⊆ Eλ+

6=cf(λ).

Theorem (Zeman, 2008). For a singular cardinal, λ:

(2λ = λ+ and �∗λ) entails ♦(S) for every S ⊆ Eλ+

cf(λ) that

reflects.

Thus, this talk will be focused on Questions 2 and 3. In

particular, we shall assume throughout that λ denotes

a singular cardinal.

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Reducing weak square

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Weak Square

Definition (Jensen ’72). �∗λ asserts the existence of a

sequence−→P = 〈Pα | α < λ+〉 such that:

1. Pα ⊆ [α]<λ and |Pα| = λ for all α < λ+;

2. for every limit γ < λ+, there exists a club

Cγ ⊆ γ satisfying:

Cγ ∩ α ∈ Pα for all α ∈ Cγ.

Remark. By Jensen, �∗λ is equivalent to the existence

of a special Aronszajn tree of height λ+.

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Approachability Property

Definition (Foreman-Magidor. implicit in Shelah ’78).

APλ asserts the existence of a seq.−→P = 〈Pα | α < λ+〉

such that:

1. Pα ⊆ [α]<λ and |Pα| = λ for all α < λ+;

2. for club many γ < λ+, there exists an unbounded

Aγ ⊆ γ satisfying:

Aγ ∩ α ∈ Pα for all α ∈ Aγ.

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Stationary Approachability Property

Definition. SAPλ asserts that for every S ⊆ Eλ+

cf(λ)

that reflects, there exists a seq.−→PS = 〈Pα | α < λ+〉

such that:

1. Pα ⊆ [α]<λ and |Pα| = λ for all α < λ+;

2. for stationarily many γ ∈ Tr(S), there exists a

stationary Sγ ⊆ S ∩ γ satisfying:

Sγ ∩ α ∈⋃{P(X) | X ∈ Pα} for all α ∈ Sγ.

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Answering question 2

Trivial Fact. �∗λ =⇒ SAPλ.

Theorem. Suppose SAPλ holds.

Then 2λ = λ+ entails ♦(S) for every S ⊆ Eλ+

cf(λ) that

reflects.

Theorem. It is relatively consistent with the existence

of a supercompact that SAPℵω holds, while �∗ℵω fails.

Moreover, SAPℵω is consistent with Refl∗([ℵω+1]ω).

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by-product: a tree from a small forcing

In one of our failed attempts to construct a model of

SAPλ+¬�∗λ, we ended-up proving the following coun-

terintuitive fact.

Theorem. It is relatively consistent with the exis-

tence of two supercompact cardinals that there exists a

cofinality-preserving forcing of size ℵ3 that introduces

a special Aronszajn tree of height ℵω1+1.

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A possible rant on our solution to Q. 2

“I do not know what SAP is, and I don’t like new defini-

tions. I know that weak square implies AP, and implies

a better scale, so why don’t you try to reduce the weak

square hypothesis from the Shelah-Zeman theorem to

these well-studied principles?”

Fortunately, we have a satisfactory response..

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In the absence of SAP

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Answering question 3

Theorem (Gitik-R.). It is relatively consistent with

the existence of a supercompact cardinal that the GCH

holds, while ♦(S) fails for some S ⊆ Eℵω+1ω that reflects.

Note that �∗ℵω necessarily fails in our model, hence, the

large cardinal hypothesis.

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More on question 3

To justify the notion of SAPλ, we also prove:

Theorem (Gitik-R.). Starting with a supercompact

cardinal, we can force to get:

(1) a strong limit λ > cf(λ) = ω with 2λ = λ+;

(2) ♦(S) fails for some S ⊆ Eλ+

cf(λ) that reflects;

in conjunction with any of the following:

• APλ+ Refl(Eλ+

cf(λ));

• a very good scale for λ;

• ∃κ < λ supercompact;

• Martin’s Maximum (so S ⊆ Eλ+ω is (ω1 + 1)-fat.)

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Revisiting the weaksquare

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Forcing axioms vs. Square, I

Magidor, extending Todorcevic, proved that PFA en-

tails the failure of �κ,ω1 for all κ > ω.

He also proved the following:

Theorem (Magidor).

(1) PFA is consistent with �∗κ for all κ;

(2) MM entails that �∗κ fails for all κ > cf(κ) = ω.

It is natural to ask whether MM can be reduced to

PFA+, in this context.

I It turns out that diamond helps..

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Forcing axioms vs. Square, II

Theorem. Suppose:

(1) λ is a singular strong limit;

(2) 2λ = λ+;

(3) �∗λ holds;

(4) every stationary subset of Eλ+

cf(λ) reflects.

then ♦∗(λ+) holds.

Remark. Replacing �∗λ with SAPλ in (3), does not yield

the conclusion! In fact, this is the approach eventually

taken to establish that SAPλ is strictly weaker than �∗λ.

Corollary. Assume PFA+.

If λ > cf(λ) = ω is a strong limit, then �∗λ fails.

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A quick proof

Corollary. Assume PFA+.

If λ > cf(λ) = ω is a strong limit, then �∗λ fails.

Proof. Suppose not. Force with Add(λ+, λ++). Then

♦∗(λ+) fails, while �∗λ and PFA+ are preserved, and λ

remains a strong limit.

It follows from the previous theorem that ♦∗(λ+)

holds. A contradiction. �

Remark. After our lecture, J. Krueger informed us of

another, already known, proof of the above corollary.

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Summary: Coherence vs. Guessing

Let Reflλ denote the assertion that every stationary

subset of Eλ+

cf(λ) reflects. Then, for λ singular, we have:

1. GCH + Reflλ+�∗λ ⇒ ♦∗(λ+);

2. GCH + Reflλ+ SAPλ 6⇒ ♦∗(λ+);

3. GCH + Reflλ+ SAPλ ⇒ ♦(S) for every stat. S ⊆ λ+;

4. GCH + Reflλ+ APλ 6⇒ ♦(S) for every stat. S ⊆ λ+.

Remark. here, the non-implication symbol, 6⇒, is a

slang for a consistency result modulo the existence of

a supercompact cardinal.

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Open problems

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Open problems

Let λ denote a singular cardinal.

Question I. Does 2λ = λ+ entail ♦(Eλ+

cf(λ))?

equivalently:

Question II. Does 2λ = λ+ entail the existence of

a stationary S ⊆ [λ+]<λ on which X 7→ sup(X) is an

injective map from S to Eλ+

cf(λ)?

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Thank you!

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